Occasions Within Phases Repeated Measures - Derivation of Expected Mean Squares

1. Introduction

Repeated measures data arise in a wide array of different research environments. The term “repeated” is used here to describe measurements that “are made on the same characteristic on the same observational unit but on more than one occasion.” (Crowder and Hand, 1990, p. 1) Because the measurements for a particular individual, subject, patient or observational unit are repeated, they are not independent within the individual. Therefore, special care must be taken in the statistical analysis of such data.

We analyze some repeated measures data taken from the textbook by Crowder and Hand (1990), Example 3.3 on page 32. Twelve hospital patients underwent a dietary regime treatment. Over the course of seven occasions, measurements were taken on ascorbic acid for each patient. The seven occasions were divided into three treatment phases - twice before, three times during, and twice after the treatment regime. The seven occasions occurred at weeks 1, 2, 6, 10, 14, 15, & 16. The ascorbic acid measurements are made on the same patient (subject) over time, and therefore, are not statistically independent within the individual patient.

Author’s Comment: This type of analysis has proven to be very useful in Cancer and AIDS research. See, for example, del Prete et al. (2014, 2015). We use this font to designate a variable, function or other object in the R statistical environment.

2. The Model

For this problem, the model may be written as

\[ y_{ijk} = \pi_{i} + \Phi_{j} + o_{k(j)} + \pi\Phi_{ij} + \pi o_{ik(j)} \]

where \(y_{ijk}\) is \(ijk^{th}\) response for the \(i^{th}\) patient on the \(k^{th}\) occasion in the \(j^{th}\) phase, \(\pi_{i}\) is the \(i^{th}\) patient, \(\Phi_{j}\) is the \(j^{th}\) phase, \(o_{k(j)}\) is the \(k^{th}\) occasion nested within the \(j^{th}\) phase, and \(\pi o_{ik(j)}\) is the interaction between the \(i^{th}\) patient and the \(k^{th}\) occasion nested within the \(j^{th}\) phase. In this analysis, the patients (patient), occasions (occ) and phases (Phase) are considered as factors in the statistical setup. In particular, the occ variable is a factor variable, not a numeric variable. Therefore, this statistical model is fully-saturated. It contains no error term. That is, there is no \(\epsilon_{ijk}\) represented in the model. The error term for testing the hypothesis of interest in the anova is constructed in a special way, as described in the sequel.

The patient, patient, represented as \(\pi_{i}\), is a random factor. A patient is randomly selected from a potentially large population of patients. However, Phase (Phase) and occ (Occasion) are fixed factors. They are measured or controlled by the investigator and are comprised of a relatively low number of finite levels. In the model, any interaction between a fixed factor and a random factor is a random factor.

In order to test for the hypothesis of whether some relative upward or downward shift occurs among Phase’s, it is necessary to construct the \(F\) statistic in a non-traditional way. To obtain the correct \(F\) statistic, it is necessary to generate the expected mean-squares from the structural model. The expected mean-squares can be derived from different algorithms, such as the Cornfield-Tukey algorithm, as outlined, for example, in Winer (1971).

3. Plots and Data Description

3.01. Accessing the ascorbic data frame from the owprm Package

The ascorbic acid data, which are used in the sequel, may be obtained from the ascorbic data frame, available through the owprm package.

install.packages("devtools") ## if not installed already
library("devtools")
devtools::install_github("wgalvord/owprm")
library(owprm)
help(package = "owprm")
data(ascorbic)

Figure 1 shows a plot of the data from Crowder and Hand (1990). This plot is constructed from a groupedData data frame from package nlme (Pinheiro & Bates, 2000). The data appear to show a rise at week 6 and a fall (drop) at week 15.

Reaction of twelve patients to treatment (Rx) across seven Occasions (occ) in three Phases.

Figure 2 shows an alternative plot of the ascorbic data, emphasizing that the occasions on the \(x\)-axis are to be considered as a factor dimension as opposed to a numeric dimension. This plot is constructed from the ggplot2 package (Wickham, 2009). [Note: In the interest of reconstructing this plot exactly, set.seed(11) before each layer of the plot produces the jittered data values positioned as seen here. {See Leek (2015), Peng (2015), Xie (2014) in connection with dynamic documentation and reproducible research}]

Reaction of twelve patients to treatment (Rx) across seven Occasions (occ) in three Phases.

On the Occasions axis, the responses are jittered to avoid over-plotting. Legends for the patients, patient, are included in the plot. For these data, we have three phases, Phase, which are designated as pre, Rx, and post to correspond to the respones before, during, and after the treatment regimen. With respect to the spread or variability of the data, it appears that the ascorbic acid responses are reasonably homogeneous throughout the seven occasions.

In these analyses, we are interested in detecting Phase shifts. The responses appear to be relatively low in the first phase, pre. They are followed by an increase in the second phase, Rx and then by a drop in the third phase, post.

4. Analysis

In this problem, there is one hypothesis of interest. It is motivated by the question: Is there a difference in the mean levels of responses across the three Phases under consideration, pre, Rx, and post? The criterion variable is ascorbic acid, i.e., ascorbic.acid in the ascorbic data frame. Here is an abbreviated anova (analysis of variance) summary table for this analysis.

##               Df Sum Sq Mean Sq
## patient       11  3.682  0.3347
## Phase          2  5.961  2.9805
## occ            4  0.203  0.0509
## patient:Phase 22  2.397  0.1090
## patient:occ   44  2.052  0.0466

The correct test for determining whether there is an effect due to Phase is constructed by taking the mean square for Phase (2.9805) and dividing it by the mean square for patient:Phase (0.1090), and entering the \(F\) distribution with the appropriate degrees of freedom, shown here. The printing of the probability value to four significant digits is deliberate.

## 
##       F (null) Num df Den df       p.val
## Value    27.36      2     22 0.000001079

The \(F\) statistic is 27.36 on 2 and 22 degrees of freedom, \(p <<\) 0.0001. Indeed, there is a significant increase in ascorbic acid responses during the treatment regimen, or Phase Rx.

For completeness, we will follow up with a test to determine whether there is a difference between the pre and post phases to determine more precisely whether the significant result is due to the increase in the Rx phase alone. As before, the question of interest is whether there is a difference in the mean levels of responses across the two phases now under consideration, pre and post?

Once again, the correct test for determining whether there is an effect due to Phase is constructed by taking the mean square for Phase and dividing it by the mean square for patient:Phase, and entering the \(F\) distribution with the appropriate degrees of freedom.

## Call: 
## owprm.default(aovObj = ch.pre.post.aov)
## 
## Anova Summary Table for Occ w/in Phases Rep Meas Analy: 
##       F (null) Num df Den df     p.val
## Value 2.060419      1     11 0.1789876
## 
## Summary of Saturated ANOVA Object: 
##                   Df Sum Sq Mean Sq
## patient           11 2.5743 0.23403
## Phase              1 0.2776 0.27755
## Phase:occ          2 0.1049 0.05244
## patient:Phase     11 1.4818 0.13471
## patient:Phase:occ 22 1.1444 0.05202

The \(F\) statistic is 2.06 on 1 and 11 degrees of freedom, \(p = 0.17899 \sim 0.18\). There is no significant difference in ascorbic acid responses between treatment phases pre and post. Across all three phases, the significant difference is due to the increase in the Rx phase alone.

Rules for Deriving the Expected Values of the Mean Squares

We now discuss the derivation of the expected mean squares. The expected mean-squares can be derived from different algorithms, such as the Cornfield-Tukey (1956) algorithm. We reproduce the discussion as presented in Winer (1971); however, we make several corrections to Winer’s presentation with respect to ‘subscripts’ being in parentheses as opposed to ‘effects’ being nested.

Section 5.14 in Winer (1971)

Given an experiment in which the underlying variables can be assumed to satisfy the conditions of the general linear model, the expected values of the mean squares computed from the experimental data can be obtained by means of a relatively simple set of rules. Although these rules lead to an end product that has been proved to be statistically correct when the assumptions underlying the general linear model are met, the rules themselves provide little insight into the mathematical rationale underlying the end product. The assumptions that underlie the general linear model have been stated in Sec. 5.6 (in Winer, 1971). The rules outlined in this section are those developed by Cornfield and Tukey (1956). A similar set of rules are also found in Bennett and Franklin (1954).

As the rules will be given, no distinction will be made between random and fixed factors. However, certain of the terms become either 0 or 1 depending upon whether an experimental veriable corresponds to a fixed or random factor. For purposes of simplifying the expressions that will result, the following notation wil be used:

\[D_{p} = 1 - \frac{p}{P}, \hspace{5 mm} i = 1, ..., p.\]

\[D_{q} = 1 - \frac{q}{Q}, \hspace{5 mm} j = 1, ..., q.\]

\[D_{r} = 1 - \frac{r}{R}, \hspace{5 mm} k = 1, ..., r.\]

If \(p = P\), that is, if factor \(\pi\) is fixed then \(D_{p}\) is zero (\(D_{p} = 1 - (1/1) = 0\)). On the other hand, if factor \(\pi\) is random, \(D_{p}\) is unity (\(D_{p} = 1 - (0/1) = 1\)). Similarly, the other \(D\)’s are either 0 or 1 depending upon whether the corresponding factor is fixed or random. In the application of these rules to designs of special interest, the appropriate evaluation of the \(D\)’s should be used rather than the \(D\)’s themselves. The general statement of the rules is followed by a series of examples.

Note: For this problem, factor \(\pi\) (patient) is random. Therefore, \(D_{p}\) is unity (\(D_{p} = 1\)). Factor \(\Phi\) (Phase) is fixed. Hence, \(D_{q}\) is zero (\(D_{q} = 0\)). Factor \(o\) (occ, Occasion) is also fixed. Therefore, \(D_{r}\) is also zero (\(D_{r} = 0\)).

Rule 1. Write the appropriate model for the design, making explicit in the notation those effects which are nested.

Rule 2. Construct a two-way table in which the terms in the model (except for the grand mean) are the row headings and the subscripts appearing in the model are the column headings. The number of columns in this table will be equal to the number of different subscripts in the model. The number of rows will be equal to the number of terms in the model that have subscripts. The row headings should include all subscripts associated with terms in the model.

Rule 3. To obtain the entries in column \(i\), enter \(D_{p}\) in those rows having headings containing an \(i\) that is not in parentheses (correcting Winer who states here ‘is not nested’); enter unity (1) in those rows having headings containing an \(i\) that is in parentheses (correcting ‘is nested’); and enter \(p\) in those rows having headings that do not contain an \(i\).

Rule 4. To obtain the entries in column \(j\), enter \(D_{q}\) in those rows having headings containing a \(j\) that is not in parentheses (correcting ‘not nested’); enter unity (1) in those rows having headings containing a \(j\) that is in parentheses (correcting ‘is nested’); and enter \(q\) in those rows having headings that do not contain a \(j\).

Rule 5. Entries in all other columns follow the general pattern outlined in rules 3 and 4. For example, the possible entries in column \(k\) would be \(D_{r}\), unity (1), and \(r\).

Rule 6. The expected value of the mean square for the main effect of factor \(\pi\) is a weighted sum of the variances due to all effects that contain the subscript \(i\). If a row heading contains a subscript \(i\), then the weight for the variance due to this row effect is the product of all entries in this row, the entry in column \(i\) being omitted. (Winer - “For nested effects, see rule 10.” Alvord - “For subscripts in parentheses see Rule 10.”)

Rule 7. The expected value of the mean square for the main effect of factor \(\Phi\) is a weighted sum of the variances due to all effects that contain the subscript \(j\). If a row heading contains a subscript \(j\), the weight for the variance due to this effect is the product of all entries in this row, the entry in column \(j\) being omitted. (Winer - For nested effects, see rule 10. Alvord for subscripts in parentheses see Rule 10.)

Rule 8. The expected value of the mean square for the \(\pi\Phi\) interaction is a weighted sum of the variances due to all effects that contain both the subscripts \(i\) and \(j\). If a row heading contains both the subscripts \(i\) and \(j\), the weight for the variance corresponding to this effect is the product of all entries in this row, the entry in columns \(i\) and \(j\) being omitted. (See Rule 10.)

Rule 9. In general, the expected value of a mean square for an effect that has the general representation \(XYZ_{uvw}\) is a weighted sum of the variances due to all effects in the model that contains all the subscripts \(u\), \(v\) and \(w\) (and possibly other subscripts). If a row heading does contain all three of the three subscripts \(u\), \(v\) and \(w\), then the weight for the variance due to the corresponding row effects is the product of all entries in this row, the entries in columns \(u\), \(v\) and \(w\) being omitted.

Rule 10. If the subscript for an effect is in parentheses (correcting Winer’s, ‘is nested’), the expected value of its mean square is a weighted sum of variances corresponding to all effects containing the same subscripts as the subscripts in parentheses (correcting Winer’s, ‘nested effect’). For example, if the main effect of factor \(B\) appears as \(\beta_{j(i)}\) in the model, then the relevant effects are those which contain both the subscripts \(i\) and \(j\). Similarly, if the term \(\beta\gamma_{j(i)k}\) appears in the model, in considering the set of relevant variances, row heading must contain all three of the subscripts \(i\), \(j\), and \(k\).

The application of these rules will be illustrated by means of \(p \times q \times r\) partially hierarchical factorial design having \(n = 1\) observation per cell. In this design, it will be assumed that factor \(o\) (for Occasions, occ) is nested under factor \(\Phi\) (for Phase, Phase). … The notation \(o_{k(j)}\) indicates that factor \(o\) (for Occasions, occ) is nested under factor \(\Phi\) (for Phase, Phase). The structural model for this design may be written as

\[ y_{ijkm} = \mu_{...} + \pi_{i} + \Phi_{j} + o_{k(j)} + \pi\Phi_{ij} + \pi o_{ik(j)} + \epsilon_{m(ijk)} \]

In accordance with Rule 1, the notation in the structural model makes explicit those effects that are nested. Thus, the notation \(o_{k(j)}\) indicates that factor \(o\) (for Occasions, occ) is nested under factor \(\Phi\) (for Phase, Phase). The notation \(\epsilon_{m(ijk)}\) indicates that the unique effects associated with an obervation on element \(m\) in \(abc_{ijk}\) is (are) nested under all effects; i.e., the experimental error is nested under all factors.

The two-way table called for by Rule 2 is given in Table 1. The row headings are the terms in the model, the term \(\mu_{...}\) being omitted. The column headings are the different subscripts that appear in the model.

Table 1 which shows the expected values of the mean squares using the \(D\) notation. The structural equation is displayed above the table.

\[ y_{ijkm} = \mu_{...} + \pi_{i} + \Phi_{j} + o_{k(j)} + \pi\Phi_{ij} + \pi o_{ik(j)} + \epsilon_{m(ijk)} \]

Effect	\(i\)	\(j\)	\(k\)	\(m\)	Expected (MS)
\(\pi_{i}\)	\(D_{p}\)	\(q\)	\(r\)	1	\(\sigma^{2}_{\epsilon} + D_{r}\sigma^2_{\pi o} + r\sigma^{2}_{\pi\Phi} + qr\sigma^{2}_{\pi}\)
\(\Phi_{j}\)	\(p\)	\(D_{q}\)	\(r\)	1	\(\sigma^{2}_{\epsilon} + \sigma^{2}_{\pi o} + r\sigma^{2}_{\pi\Phi} + D_{r}p\sigma^{2}_{o} + pr\sigma^{2}_{\Phi}\)
\(o_{k(j)}\)	\(p\)	\(1\)	\(D_{r}\)	1	\(\sigma^{2}_{\epsilon} + \sigma^{2}_{\pi o} + p\sigma^{2}_{o}\)
\(\pi\Phi_{ij}\)	\(D_{p}\)	\(D_{q}\)	\(r\)	1	\(\sigma^{2}_{\epsilon} + \sigma^{2}_{\pi o} + r\sigma^{2}_{\pi\Phi}\)
\(\pi o_{ik(j)}\)	\(D_{p}\)	\(1\)	\(D_{r}\)	1	\(\sigma^{2}_{\epsilon} + \sigma^{2}_{\pi o}\)
\(\epsilon_{m(ijk)}\)	1	1	1	1	\(\sigma^{2}_{\epsilon}\)

As stated previously, for this problem, factor \(\pi\) is random. Therefore, \(D_{p}\) is unity (\(D_{p} = 1\)). Factor \(\Phi\) is fixed. Therefore, \(D_{q}\) is zero (\(D_{q} = 0\)). Factor \(o\) is also fixed. Therefore, \(D_{r}\) is zero (\(D_{r} = 0\)). In this next table we substitute the values of 0 or 1 for the \(D\)’s.

Table 2 shows the expected values of the mean squares in which the \(D\)’s are replaces with 0’s or 1’s. The structural equation is, again, displayed above the table.

\[ y_{ijkm} = \mu_{...} + \pi_{i} + \Phi_{j} + o_{k(j)} + \pi\Phi_{ij} + \pi o_{ik(j)} + \epsilon_{m(ijk)} \]

Effect	\(i\)	\(j\)	\(k\)	\(m\)	Expected (MS)
\(\pi_{i}\)	1	\(q\)	\(r\)	1	\(\sigma^{2}_{\epsilon} + r\sigma^{2}_{\pi\Phi} + qr\sigma^{2}_{\pi}\)
\(\Phi_{j}\)	\(p\)	0	\(r\)	1	\(\sigma^{2}_{\epsilon} + r\sigma^{2}_{\pi\Phi} + pr\sigma^{2}_{\Phi}\)
\(o_{k(j)}\)	\(p\)	\(1\)	0	1	\(\sigma^{2}_{\epsilon} + \sigma^{2}_{\pi o} + p\sigma^{2}_{o}\)
\(\pi\Phi_{ij}\)	1	0	\(r\)	1	\(\sigma^{2}_{\epsilon} + r\sigma^{2}_{\pi\Phi}\)
\(\pi o_{ik(j)}\)	1	\(1\)	0	1	\(\sigma^{2}_{\epsilon} + \sigma^{2}_{\pi o}\)
\(\epsilon_{m(ijk)}\)	1	1	1	1	\(\sigma^{2}_{\epsilon}\)

In Table 2 the expected mean squares are more easily interpreted than they are in Table 1. In row 2 of Table 2 (Effect: \(\Phi_{j}\)) the expected mean square is \(\sigma^{2}_{\epsilon} + r\sigma^{2}_{\pi\Phi} + pr\sigma^{2}_{\Phi}\). In row 4 of Table 2 (Effect: \(\pi\Phi_{ij}\)) the expected mean square is \(\sigma^{2}_{\epsilon} + r\sigma^{2}_{\pi\Phi}\). These expected mean squares differ by the quantity \(pr\sigma^{2}_{\Phi}\). For our problem, the correct test for determining whether there is an effect due to phases is constructed by taking the mean square for Phase and dividing it by the mean square for patient:Phase, and entering the \(F\) distribution with the appropriate degrees of freedom. Thus, if the variance due to phases, \(\sigma^{2}_{\Phi}\) is statistically equal to zero, the \(F\) statistic will be ‘small’ and the effect due to phases, Phase, will be non significant. On the other hand, if the variance due to phases is greater than zero, the \(F\) statistic will be ‘large’ and the effect due to phases will be significant.

Additional section extracted from a pdf on the Web

Here is a portion of an alternative set of rules that was extracted from a pdf on the Web at this site:

www.fc.up.pt/pessoas/amsantos/pdfs/ct-rules.pdf

Table 3 shows a similar table to Tables 1 and 2, but with type of variable, fixed or random, and the numbers of levels shown at the top of the table.

Rules:

For each column (starting with \(i\), say), start at the bottom (and move up):

1. Subscript Not Present If the subscript for that column is not present, insert the levels associated with the subscript. This will be \(p\) for column \(i\), \(q\) for columnn \(j\), and \(r\) for column \(k\).

2. Subscript IS Present (2a) Subscript present and in parentheses: If the subscript for that particular column is present, and the subscript is in parentheses, insert a 1. (2b) Subscript present and not in parentheses and factor is random: If the subscript for that particular column is present, and the subscript is NOT in parentheses, insert a 1 if the factor is random. (2c) Subscript present and not in parentheses and factor is fixed: If the subscript for that particular column is present, and the subscript is NOT in parentheses, insert a 0 if the factor is fixed.

\[ y_{ijkm} = \mu_{...} + \pi_{i} + \Phi_{j} + o_{k(j)} + \pi\Phi_{ij} + \pi o_{ik(j)} + \epsilon_{m(ijk)} \]

Type	R	F	F	R	Expected (MS)
Levels	\(p=12\)	\(q=3\)	\(r=7\)	\(n=1\)	Expected (MS)
Effect	\(i\)	\(j\)	\(k\)	\(m\)	Expected (MS)
———	———	——–	——–	——–	———————–
\(\pi_{i}\)	1	\(q\)	\(r\)	1	\(\sigma^{2}_{\epsilon} + r\sigma^{2}_{\pi\Phi} + qr\sigma^{2}_{\pi}\)
\(\Phi_{j}\)	\(p\)	0	\(r\)	1	\(\sigma^{2}_{\epsilon} + r\sigma^{2}_{\pi\Phi} + pr\sigma^{2}_{\Phi}\)
\(o_{k(j)}\)	\(p\)	\(1\)	0	1	\(\sigma^{2}_{\epsilon} + \sigma^{2}_{\pi o} + p\sigma^{2}_{o}\)
\(\pi\Phi_{ij}\)	1	0	\(r\)	1	\(\sigma^{2}_{\epsilon} + r\sigma^{2}_{\pi\Phi}\)
\(\pi o_{ik(j)}\)	1	\(1\)	0	1	\(\sigma^{2}_{\epsilon} + \sigma^{2}_{\pi o}\)
\(\epsilon_{m(ijk)}\)	1	1	1	1	\(\sigma^{2}_{\epsilon}\)

References

Alvord WG (2016) Introduction to owprm - Occasions within Phases Repeated Measures, https://RPubs.com/wgalvord

Bennett CA and Franklin NL (1954) Statistical Analysis in Chemistry and the Chemical Insustry, Wiley.

Cornfield J and Tukey JW (1956) ‘Average values of mean squares in factorials’ Annals of Mathematical Statistics, 27, 907-949.

Crowder MJ and Hand DJ (1990) Analysis of Repeated Measures, Chapman and Hall.

Del Prete GQ, Oswald K, Lara A, Shoemaker R, Smedley J, Macallister R, Coalter V, Wiles A, Wiles R, Li Y, Fast R, Kaiser R, Lu B, Zheng J, Alvord WG, Trubey CM, Piatak M, Deleage C, Keele BF, Estes JD, Hesselgesser J, Geleziunas R, Lifson JD (2015) “Elevated plasma viral loads in romidepsin treated SIV-infected rhesus macaques on suppressive combination antiretroviral therapy.” Antimicrobial Agents and Chemotherapy, 60: 1560-1572, doi:10.1128/AAC.02625-15.

Del Prete GQ, Shoemaker R, Oswald K, Lara A, Trubey CM, Fast R, Schneider DK, Kiser R, Coalter V, Wiles A, Wiles R, Freemire B, Keele BF, Estes JD, Quiñones OA, Smedley J, Macallister R, Sanchez RI, Waid JS, Tane CM, Alvord WG, Hazuda DJ, Piatak M, Lifson JD “Effect of suberoylanilide hydroxamic acid (SAHA) administration on the residual virus pool in a model of combination antiretroviral therapy-mediated suppression in SIVmac239-infected indian rhesus macaques.” Antimicrobial Agents and Chemotherapy, 58: 6790-806. PMID 25182644 DOI: 10.1128/AAC.03746-14

Leek J (2015) The Elements of Data Analytic Style, Leanpub.

Peng R (2015) Report Writing for Data Science in R, Leanpub.

Winer BJ (1971) Statistical Principles in Experimental Design, 2nd ed., McGraw-Hill.

Xie Y (2014) Dynamic Documents with R and knitr, CRC Press.

Keywords

repeated measures, occasions, phases, Cornfield-Tukey algorithm, expected mean squares

The ascorbic acid responses in table form

In this section we display the Ascorbic Acid, ascorbic.acid, responses for each of the 12 patients, patient, over the 7 occasions, Occasion.

##         
##             1    2    3    4    5    6    7
##   Pat.01 0.22 0.00 1.03 0.67 0.75 0.65 0.59
##   Pat.02 0.18 0.00 0.96 0.96 0.98 1.03 0.70
##   Pat.07 0.30 1.09 1.17 0.90 1.17 0.75 0.88
##   Pat.03 0.73 0.37 1.18 0.76 1.07 0.80 1.10
##   Pat.09 0.31 0.54 1.24 0.56 0.77 0.28 0.40
##   Pat.12 0.73 0.50 1.08 1.26 1.17 0.91 0.87
##   Pat.11 0.60 0.80 1.02 1.28 1.16 1.01 0.67
##   Pat.05 0.54 0.42 1.33 1.32 1.30 0.74 0.56
##   Pat.06 0.16 0.30 1.27 1.06 1.39 0.63 0.40
##   Pat.04 0.30 0.25 0.74 1.10 1.48 0.39 0.36
##   Pat.10 1.40 1.40 1.64 1.28 1.12 0.66 0.77
##   Pat.08 0.70 1.30 1.80 1.80 1.60 1.23 0.41